What is the integral from 1 to e^2 of ln(sqrt(x)) ?

The integral from 1 to e^2 of ln(sqrt(x)) is (e^2+1)/2

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integral from 1 to e^2 of ln(sqrt(x))

\int_{1}^{e^{2}} ln (x) d x

\frac{e ^{2} + 1}{2}

Decimal Notation

4.19452 \dots

Solution steps

\int_{1}^{e^{2}} ln (x) d x

Apply Integration By Parts

: [x ln (x) - \int \frac{1}{2} d x]_{1}^{e^{2}}

= [x ln (x) - \int \frac{1}{2} d x]_{1}^{e^{2}}

\int \frac{1}{2} d x = \frac{1}{2} x

= [x ln (x) - \frac{1}{2} x]_{1}^{e^{2}}

Simplify

[x ln (x) - \frac{1}{2} x]_{1}^{e^{2}} : [\frac{1}{2} (x ln (x) - x)]_{1}^{e^{2}}

= [\frac{1}{2} (x ln (x) - x)]_{1}^{e^{2}}

Compute the boundaries:

\frac{e ^{2}}{2} + \frac{1}{2}

= \frac{e ^{2}}{2} + \frac{1}{2}

Simplify

= \frac{e ^{2} + 1}{2}

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What is the integral from 1 to e^2 of ln(sqrt(x)) ?
The integral from 1 to e^2 of ln(sqrt(x)) is (e^2+1)/2